Mathematical Science: Analytical and Topological Invariants of Odd-Dimensional Manifolds and Cell-Complexes

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9401516 Klassen Eric Klassen and Paul Kirk intend to develop powerful tools to compute and analyze gauge-theoretic invariants of 3-dimensional manifolds, particularly Chern-Simons invariants and the spectral flow of the signature operator, with an emphasis on manifolds with boundary. The techniques to be used include the theory of boundary-value problems for Dirac operators, Massey products in differential graded Lie algebras, and the representation theory of fundamental groups of 3-manifolds. Over the last decade, gauge theoretic topology has grown into one of the most important current fields of mathematical research. It involves an interaction of ideas from topology, analysis, differential and algebraic geometry, and theoretical physics. The exciting techniques resulting from this interaction have been used to answer fundamental questions about 3- and 4-dimensional manifolds which had been completely resistant to pre-gauge theory methods. The research to be undertaken forms one of the basic ingredients of both the 3- and 4-dimensional theories; it will provide information important for cut-and-paste methods in the 4-dimensional Donaldson theory, as well as providing computations for 3-dimensional topological quantum field theory. ***